Method and apparatus for adaptive gravity search

ABSTRACT

Provided are a gravity search adaptive apparatus and method. The gravity search adaptive apparatus includes: a system configured to process and output a received signal; a system modeling unit configured to receive the same signal as a signal input to the system, to convert the signal using a system modeling parameter, and to output the converted signal; and an adaptive controller configured to use a gravity search adaptive algorithm to detect the system modeling parameter so that an error signal which is a difference between an output signal of the system and an output signal of the system modeling unit converges on a minimum value when the system modeling parameter is applied to the system modeling unit.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit under 35 U.S.C. §119(a) of KoreanPatent Applications No. 10-2010-0133789, filed on Dec. 23, 2010, and No.10-2011-0090383, filed on Sep. 6, 2011, the entire disclosures of whichare incorporated herein by reference for all purposes.

BACKGROUND

1. Field

The following description relates to system modeling, and moreparticularly, to an apparatus and method for adaptively modeling asystem by estimating system model parameters using statisticalcharacteristics of signals.

2. Description of the Related Art

A signal processing system needs an algorithm for optimizing itselfunder a predetermined criterion upon signal processing when noinformation about a certain input signal is given, in order to varysystem characteristics as necessary. Such an algorithm is called anadaptive algorithm.

A representative adaptive algorithm is a steepest descent algorithm.However, the steepest descent algorithm has a problem that it is noteasy to adjust a convergence speed.

SUMMARY

The following description relates to an apparatus and method foradaptive gravity search, capable of easily adjusting a convergence speedand a convergence pattern.

In one general aspect, there is provided a gravity search adaptiveapparatus for modeling a system, including: a system modeling unitconfigured to receive the same signal as a signal input to the system,to convert the signal using a system modeling parameter, and to outputthe converted signal; and an adaptive controller configured to use agravity search adaptive algorithm to detect the system modelingparameter so that an error signal which is a difference between anoutput signal of the system and an output signal of the system modelingunit converges on a minimum value.

In another general aspect, there is provided a gravity search adaptivemethod including: calculating a system modeling parameter of a systemmodeling unit for minimizing an error signal which is a differencebetween an output signal of a system and an output signal of the systemmodeling unit when the same signal has been input to the system and thesystem modeling unit; and applying the system modeling parameter to thesystem modeling unit.

Other features and aspects will be apparent from the following detaileddescription, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram illustrating a configuration example of a systemmodeling apparatus for implementing a general adaptive algorithm.

FIGS. 2A and 2B are views for explaining a convergence speed when asteepest descent algorithm is applied.

FIG. 3 is a view for explaining a principle of a gravity search adaptivealgorithm.

FIG. 4 is a view for explaining a relationship between ε and a parameterwhen a single parameter is used.

FIG. 5 is a view for explaining power components that act on a ball.

FIGS. 6A through 6C are views for explaining an acceleration accordingto a change in movement direction of a ball.

FIG. 7A shows a gradient vector

[k].

FIG. 7B shows a velocity vector

[k].

FIG. 8 shows the curvature of a curve.

FIG. 9 is a view for explaining an acceleration when the movementdirection of a ball changes.

FIG. 10 is a diagram illustrating an example of a gravity searchadaptive apparatus.

FIG. 11 is a flowchart illustrating an example of a system modelingmethod using a gravity search adaptive algorithm.

Throughout the drawings and the detailed description, unless otherwisedescribed, the same drawing reference numerals will be understood torefer to the same elements, features, and structures. The relative sizeand depiction of these elements may be exaggerated for clarity,illustration, and convenience.

DETAILED DESCRIPTION

The following description is provided to assist the reader in gaining acomprehensive understanding of the methods, apparatuses, and/or systemsdescribed herein. Accordingly, various changes, modifications, andequivalents of the methods, apparatuses, and/or systems described hereinwill be suggested to those of ordinary skill in the art. Also,descriptions of well-known functions and constructions may be omittedfor increased clarity and conciseness.

FIG. 1 is a diagram illustrating a configuration example of a systemmodeling apparatus for implementing a general adaptive algorithm.

Referring to FIG. 1, the system modeling apparatus includes a system110, a system modeling unit 120, and an adaptive controller 130.

The system 110 processes and outputs input signals. The processingcharacteristics of the system 110 are not known or changes in timedepend on environmental factors.

In FIG. 1, x[k] represents an input signal of the system 110, which is a(N×1) vector, and y[k] represents an output signal from the system 110.

The system modeling unit 120 has been modeled as mathematicalrepresentation of the system 110. The system modeling unit 120 is usedto model a unknown system or a system whose characteristics varydepending on environmental factors. {right arrow over (w)}[k], which isan (N×1) vector, represents a parameter vector of the system modelingunit 120. The system modeling unit 120 receives the input signal x[k] ofthe system 110 and applies {right arrow over (w)}[k] to the input signalx[k], thus outputting an output signal ŷ[k].

The adaptive controller 130 controls the system 110 and the systemmodeling unit 120 so that the same signal is applied to the system 110and the system modeling unit 120, and compares an output signal from thesystem 110 to an output signal from the system modeling unit 120 tothereby estimate a parameter of the system modeling unit 120. In detail,the adaptive controller 130 includes an error detector 131 and amodeling parameter calculator 132.

The error detector 131 detects a difference between the output signaly[k] of the system 110 and the output signal ŷ[k] of the system modelingunit 120, thus outputting the detected difference as an error signale[k]. Then, the modeling parameter calculator 132 calculates a modelingparameter vector for reducing power of the error signal e[k] output fromthe error detector 131, and applies the modeling parameter vector to thesystem modeling unit 120. Then, the modeling parameter calculator 132continues to calculate modeling parameters until the error signal e[k]between the output signal y[k] of the system 110 and the output signalŷ[k] of the system modeling unit 120 obtained when the previouslycalculated modeling parameter is applied to the system modeling unit 120converges on a minimum value, and supplies the calculated modelingparameters to the system modeling unit 120 to thereby update the systemmodeling unit 120.

That is, the adaptive controller 130 estimates a system modelingparameter for approximating the output value of the system modeling unit120 to the output value y[k] of the system 110, thereby estimating thecharacteristics of the system 110.

The output signal ŷ[k] of the system modeling unit 120 at time k isexpressed by equation 1 below.

$\begin{matrix}{{\hat{y}\lbrack k\rbrack} = {{\sum\limits_{n = 0}^{N - 1}{{w_{k}^{*}\lbrack k\rbrack} \cdot {x\left\lbrack {k - n} \right\rbrack}}} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}}}} & (1)\end{matrix}$

In equation 1, H represents Hermitian Transpose, and (N×1) vectors{right arrow over (w)}[k] and {right arrow over (x)}[k] are defined asfollows.

${{\overset{\rightharpoonup}{w}\lbrack k\rbrack} = \begin{bmatrix}{w_{0}\lbrack k\rbrack} \\{w_{1}\lbrack k\rbrack} \\\vdots \\{w_{N - 1}\lbrack k\rbrack}\end{bmatrix}},\mspace{14mu}{{\overset{\rightharpoonup}{x}\lbrack k\rbrack} = \begin{bmatrix}{x\lbrack k\rbrack} \\{x\left\lbrack {k - 1} \right\rbrack} \\\vdots \\{x\left\lbrack {k - N + 1} \right\rbrack}\end{bmatrix}}$

An adaptive algorithm is aimed at obtaining a modeling parameter vectorfor minimizing the power of an error signal which is a differencebetween the output signal of a real system and the output signal of asystem modeling unit.

When noise n[k] exists in a measured signal, the error signal e[k] canbe expressed by equation 2 below.e[k]=y[k]−ŷ[k]+n[k]  (2)

Also, the power ε of the error signal e[k] is given as equation 3 below.

$\begin{matrix}\begin{matrix}{ɛ = {{E\left\{ {{e\lbrack k\rbrack}}^{2} \right\}} = {E\left\{ {{e\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} \right\}}}} \\{= {E\left\{ {{{y\lbrack k\rbrack}}^{2} - {{y\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{w}\lbrack k\rbrack}} - {{\overset{\rightharpoonup}{w}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{y^{*}\lbrack k\rbrack}} +} \right.}} \\{{{\overset{\rightharpoonup}{w}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{w}\lbrack k\rbrack}} + {{y\lbrack k\rbrack}{n^{*}\lbrack k\rbrack}} + {{n\lbrack k\rbrack}{y^{*}\lbrack k\rbrack}} -} \\\left. {{{\overset{\rightharpoonup}{w}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{n^{*}\lbrack k\rbrack}} - {{n\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{w}\lbrack k\rbrack}} + {{n^{*}\lbrack k\rbrack}{n\lbrack k\rbrack}}} \right\} \\{{= {\sigma_{y}^{2} - {{\overset{\rightharpoonup}{p}}^{H}{\overset{\rightharpoonup}{w}\lbrack k\rbrack}} - {{\overset{\rightharpoonup}{w}\lbrack k\rbrack}^{H}\overset{\rightharpoonup}{p}} + {{\overset{\rightharpoonup}{w}\lbrack k\rbrack}^{H}R\;{\overset{\rightharpoonup}{w}\lbrack k\rbrack}} + \sigma_{n}^{2}}},}\end{matrix} & (3)\end{matrix}$

In equation 3, E{•} represents an ensemble average.

Also, R, which is a (N×N) matrix, can be expressed by equation 4 below.R=E{

[k]

[k] ^(H)}  (4)

In equation 3, {right arrow over (P)}, which is a (N×1) vector, can bedefined as equation 5 below.

=E{

[k]y*[k]}  (5)

In equation 3, ε is also called a cost function, and since the costfunction ε is a quadratic function of {right arrow over (w)}[k], anoptimum coefficient vector {right arrow over (w)}_(opt) at which thegradient of ε becomes zero so that c reaches a minimum value can beobtained. If {right arrow over (∇)}[k] is a gradient vector of ε, {rightarrow over (∇)}[k] can be expressed by equation 6 below.

$\begin{matrix}{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack} = {{2\frac{\partial ɛ}{\partial{\overset{\rightharpoonup}{w}\lbrack k\rbrack}^{*}}} = {{{- 2}\overset{\rightharpoonup}{p}} + {2\; R\;{\overset{\rightharpoonup}{w}\lbrack k\rbrack}}}}} & (6)\end{matrix}$

If the inverse matrix R⁻¹ of R exists, the optimum coefficient vector{right arrow over (w)}_(opt) and the optimum power ε_(opt) of the errorsignal ε when {right arrow over (w)}[k]={right arrow over (w)}_(opt) aregiven as equations 7 and 8 below.

_(opt) =R ⁻¹

  (7)ε_(opt)=σ_(y) ² −

R ⁻¹

+σ_(n) ²  (8)

However, since directly calculating the inverse matrix R⁻¹ in equation 7requires a large amount of calculation, applying a method of directlyobtaining the inverse matrix R⁻¹ in implementing a real system isimproper in many cases. In those cases, an adaptive algorithm whichrepeatedly updates {right arrow over (w)}[k] so that {right arrow over(w)}[k] gradually converges on {right arrow over (w)}_(opt) is utilized,instead of directly calculating the inverse matrix R⁻¹. One of adaptivealgorithms that has been widely used for the purpose is a steepestdescent algorithm, and the steepest descent algorithm is also called agradient descent algorithm.

The steepest descent algorithm calculates a gradient vector {right arrowover (∇)}[k] of ε with respect to the parameter vector {right arrow over(w)}[k] using the fact that ε is a quadratic function of the parametervector {right arrow over (w)}[k], and updates the parameter vector{right arrow over (w)}[k] by the magnitude of the calculated gradientvector {right arrow over (∇)}[k] in the opposite direction of thegradient to thereby obtain a next parameter vector Δ{right arrow over(w)}[k+1].

By repeating the process, the parameter vector {right arrow over (w)}[k]finally converges on the optimum coefficient vector {right arrow over(w)}_(opt) for minimizing ε as the k value increases.

The steepest descent algorithm can be expressed by equation 9 below.

$\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{{\overset{\rightharpoonup}{w}\lbrack k\rbrack} - {\frac{\mu}{2}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} - {\mu\left( {{R\;{\overset{\rightharpoonup}{w}\lbrack k\rbrack}} - \overset{\rightharpoonup}{p}} \right)}}}} & (9)\end{matrix}$

A convergence speed of the steepest descent algorithm can be adjustedusing a μ value. That is, the μ value, which is a proportionalcoefficient used to update the optimum coefficient vector {right arrowover (w)}_(opt), decides the convergence speed. If the μ value is large,the convergence speed is fast, and if the μ value is small, theconvergence speed is slow.

Meanwhile, if it is defined that Δ{right arrow over (w)}[k]={right arrowover (w)}[k]−{right arrow over (w)}_(opt), equations 10 and 11 can beinduced from equations 7 and 8.Δ

[k+1]=(I−μR)Δ

[k]=(I−μR)^(k+1)Δ

[0]  (10)ε=ε_(opt) +Δ

[k] ^(H) RΔ

[k]  (11)

If R=VAV⁻¹ is given as eigen-value decomposition of R, Δ{right arrowover (w)}[k+1] can be rewritten as equation 12 below.Δ

[k+1]=V(I−μΛ)^(k−1) V ⁻¹Δ

[0]  (12)

As seen from equation 12, in order to make the steepest descentalgorithm converge, the value has to satisfy equation 13 below.0<μ<1/λ_(max)  (13)

In order to implement the steepest descent algorithm written as equation9, the matrix R and the vector {right arrow over (P)} defined inequations 4 and 5 respectively have to be given. One method is tocalculate time averages of samples and use the time averages asestimated values of R and {right arrow over (P)}, instead of directlyobtaining R and {right arrow over (P)} using an ensemble average. Awell-known Least Mean Squares (LMS) algorithm uses a single sample ofeach of {right arrow over (x)}[k] and y[k] in order to estimate R and{right arrow over (P)}, as written by equations 14 and 15 below.{circumflex over (R)}=

[k]

[k] ^(H)  (14)

=

[k]y*[k]  (15)

As such, by estimating the instantaneous values of R and {right arrowover (P)}, the steepest descent algorithm is simplified to a LSMalgorithm, which is expressed as equation 16 below.

[k+1]=

[k]+μ

[k]e*[k]  (16)

The convergence speed of the LSM algorithm may be adjusted using the μvalue. As seen in equation 13, as the eigen-value is large, a small μvalue may be used, and if the eigen-value is small, a large μ value maybe used.

FIGS. 2A and 2B are views for explaining a convergence speed when thesteepest descent algorithm is applied.

FIGS. 2A and 2B show curves of ε when two signals having differentcharacteristics are input. However, since the μ value has to be smallerthan 1/λ_(max), if the ratio λ_(max)/λ_(min) of the maximum eigen-valuewith respect to the minimum eigen-value is large, the selected μ valuemay be too small in view of the minimum eigen-value, which may make theconvergence speed of the LMS algorithm slow. In addition to such aconvergence speed problem, the LMS algorithm may be subject toperformance deterioration due to the influence by noise when noise ismixed in a measured signal.

Accordingly, in order to overcome the problem of the steepest descentalgorithm, gravity is used. Since a new adaptive algorithm proposed inthis specification has been developed based on physical characteristicsthat occur as a natural phenomenon, it has a stable characteristic bythe law of energy conservation. Also, since the proposed adaptivealgorithm has degrees of freedom two in adjusting the convergence speedof the adaptive algorithm, the adaptive algorithm can adjust theconvergence speed and the convergence pattern more flexibly according toapplication fields.

FIG. 3 is a view for explaining a principle of a gravity search adaptivealgorithm. FIG. 3 shows an example where a ball moves in a parabolic,concave container filled up with liquid, wherein m represents the massof the ball and g represents the gravitational acceleration.

Since gravity pulls down the ball, the ball moves downward to the bottomof the container along the inner surface of the container. At this time,the potential energy of the ball is converted to kinetic energy to movethe ball, the kinetic energy is lost due to friction with the liquid,and resultantly the ball is stopped at the lowest position (that is, thebottom of the container) of the container.

At this time, the behavior of the ball, that is, how fast the ball movestoward the bottom of the container or how quickly the ball is stopped atthe bottom depends on the mass m of the ball, the magnitude g ofgravitational acceleration g, and the friction force f of the liquid.

As such, a natural phenomenon can be used as a model to obtain anadaptive algorithm for solving a problem of finding a global minimum ofa cost function which is in the form of a quadratic function. Theadaptive algorithm is called a “gravity search adaptive algorithm”, andthe gravity search adaptive algorithm is stable and has a low risk ofdivergence compared to other adaptive algorithms that use no physicalphenomenon associated with the law of energy conservation.

For easy understanding, a procedure of inducing a 1-dimensional gravitysearch adaptive algorithm whose coefficients are real numbers, aprocedure of inducing a N-dimensional gravity search adaptive algorithmwhose coefficients are real numbers, and a procedure of inducing aN-dimensional gravity search adaptive algorithm whose coefficients arecomplex numbers will be described in this order below.

<1-Dimensional Gravity Search Adaptive Algorithm>

In the case of a 1-dimensional gravity search adaptive algorithm, thecost function of equation 3 can be simplified to equation 17 below.ε=σ_(y) ²−2pw[k]+σ _(x) ² w[k] ²  (17)where w[k] is a value resulting from k-th iteration of the coefficientw, p=E{x[k]y[k]}, and σ_(x)=E{x²[k]}.

First-order and second-order derivatives of the cost function withrespect to w[k] are written as equations 18 and 19 below.

$\begin{matrix}{\frac{\mathbb{d}ɛ}{\mathbb{d}{w\lbrack k\rbrack}} = {{\nabla\lbrack k\rbrack} = {{2\;\sigma_{x}^{2}{w\lbrack k\rbrack}} - {2\; p}}}} & (18) \\{\frac{\mathbb{d}^{2}ɛ}{\mathbb{d}{w\lbrack k\rbrack}^{2}} = {2\;\sigma_{x}^{2}}} & (19)\end{matrix}$

FIG. 4 is a view for explaining a relationship between c and a parameterwhen a single parameter is used. The following description will be givenwith reference to FIGS. 3 and 4.

In FIG. 4, the w-axis represents the parameter, and the y-axisrepresents ε. That is, ε is the second-order function of w[k], and thepresent invention is aimed at obtaining w[k]=w_(opt) for minimizing ε.The w-axis component of movement of a ball, which represents theposition of the ball, can be represented as a second-order differentialequation written by equation 20 below.m{umlaut over (w)}+f{dot over (w)}=F  (20)where F represent a force applied to the ball in the direction of thew-axis, m represents the mass of the ball, f represents a friction forcedepending on viscosity of liquid, and {dot over (w)} and {umlaut over(w)} are first-order and second-order derivatives with respect to time,respectively. The F value depends on the magnitude g of gravitationacceleration, the shape of the container, and the position and movementof the ball.

If the F value is assumed to be a constant and v={dot over (w)},equation 20 can be rewritten to equation 21 below.

$\begin{matrix}{\overset{.}{v} = {{{- \frac{f}{m}}v} + \frac{F}{m}}} & (21)\end{matrix}$

A solution of the differential equation of equation 21 can be expressedas equation 22 below.

$\begin{matrix}{{v(t)} = {{{v\left( t_{0} \right)}{\mathbb{e}}^{{- \frac{f}{m}}{({t - t_{0}})}}} + {\frac{F}{m}{\int_{t_{0}}^{t}{{\mathbb{e}}^{{- \frac{f}{m}}{({t - \sigma})}}{\mathbb{d}\sigma}}}}}} & (22)\end{matrix}$

By calculating the second term of the right side of equation 22,equation 22 can be arranged to equation 23 below.

$\begin{matrix}{{v(t)} = {{{v\left( t_{0} \right)}{\mathbb{e}}^{{- \frac{f}{m}}{({t - t_{0}})}}} + {\frac{F}{f}\left( {1 - {\mathbb{e}}^{{- \frac{f}{m}}{({t - t_{0}})}}} \right)}}} & (23)\end{matrix}$

In equation 23, w(t) can be obtained as equation 24 below. v(t) is aderivative of w(t). So, if we know v(t), w(t) can be calculated bycalculating the integral of v(t).

$\begin{matrix}\begin{matrix}{{w(t)} = {{w\left( t_{0} \right)} + {\int_{t_{0}}^{t}{{v(\tau)}{\mathbb{d}\tau}}}}} \\{= {{w\left( t_{0} \right)} + {\frac{F}{f}\left( {t - t_{0}} \right)} + {\left( {{v\left( t_{0} \right)} - \frac{F}{f}} \right)\frac{m}{f}\left( {1 - {\mathbb{e}}^{{- \frac{f}{m}}{({t - t_{0}})}}} \right)}}}\end{matrix} & (24)\end{matrix}$

If First-order Taylor Series Approximation is used,

${\mathbb{e}}^{x} = {{1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \cdots} \approx {1 + x}}$when |x|<<1, and accordingly, if

${{{\frac{f}{m}\left( {t - t_{0}} \right)}} ⪡ 1},$equations 23 and 24 can be rewritten to equation 25 and 26 below.

$\begin{matrix}{{v(t)} = {{{{v\left( t_{0} \right)}{\mathbb{e}}^{{- \frac{f}{m}}{({t - t_{0}})}}} + {\frac{F}{f}\left( {1 - {\mathbb{e}}^{{- \frac{f}{m}}{({t - t_{0}})}}} \right)}} \approx {{{v\left( t_{0} \right)}\left( {1 - {\frac{f}{m}\left( {t - t_{0}} \right)}} \right)} + {\frac{F}{m}\left( {t - t_{0}} \right)}}}} & (25) \\{{w(t)} = {{{w\left( t_{0} \right)} + {\frac{F}{f}\left( {t - t_{0}} \right)} + {\left( {{v\left( t_{0} \right)} - \frac{F}{f}} \right)\frac{m}{f}\left( {1 - {\mathbb{e}}^{{- \frac{f}{m}}{({t - t_{0}})}}} \right)}} \approx {{w\left( t_{0} \right)} + {{v\left( t_{0} \right)}\left( {t - t_{0}} \right)}}}} & (26)\end{matrix}$

In equation 24, if it is assumed that v(τ) is a constant in the integralinterval, equation 24 can be rewritten to equation 27 below.w(t)≈w(t ₀)+v(t ₀)(t−t ₀)≈w(t ₀)+v(t)(t−t ₀)  (27)

In this case, equation 27 becomes identical to equation 26. This meansthat the First-order Taylor Series Approximation has the same effect asthe assumption that v(τ) is a constant. Also, since equation 27 has beenobtained under the same assumption that v(τ) is a constant, anapproximate value having the same accuracy can be obtained. In thisspecification, equation 27 is used.

Equations 25 and 27 can be arranged and expressed as matrix equation 28below.

$\begin{matrix}{\begin{bmatrix}{w(t)} & {v(t)}\end{bmatrix} = {\begin{bmatrix}{w\left( t_{0} \right)} & {v\left( t_{0} \right)}\end{bmatrix}{\quad{\begin{bmatrix}1 & 0 \\{1 - {\frac{f}{m}\left( {t - t_{0}} \right)}} & {1 - {\frac{f}{m}\left( {t - t_{0}} \right)}}\end{bmatrix}{\quad{+ {\quad\left\lbrack \begin{matrix}{\frac{F}{m}\left( {t - t_{0}} \right)} & \left. {\frac{F}{m}\left( {t - t_{0}} \right)} \right\rbrack\end{matrix} \right.}}}}}}} & (28)\end{matrix}$

Here, w(t) can be substituted by w[k+1], v(t) can be substituted byv[k+1], w(t₀) can be substituted by w[k], v(t₀) can be substituted byv[k], and F can be substituted by F[k], and t−t₀=T_(s).

Ts can be considered as a sampling interval and can be, for convenienceof expansion, normalized to Ts=1.

When the substitutions are applied, equation 28 can be rewritten toequation 29 below.

$\begin{matrix}\left\lbrack {\begin{matrix}{w\left\lbrack {k + 1} \right\rbrack} & {v\left\lbrack {k + 1} \right\rbrack}\end{matrix} = \left\lbrack {{\begin{matrix}{w\lbrack k\rbrack} & \left. {v\lbrack k\rbrack} \right\rbrack\end{matrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {\frac{F\lbrack k\rbrack}{m}\begin{bmatrix}1 & 1\end{bmatrix}}} \right.} \right. & (29)\end{matrix}$

Here, since

${\beta = {{1 - {\frac{f}{m}\mspace{14mu}{and}\mspace{14mu}\frac{f}{m}\left( {t - t_{0}} \right)}} ⪡ 1}},$β has to satisfy equation 30 below.1−ε<β<1,ε<<1  (30)

In equation 29, F[k] consists of two components as written in equation31 below.F[k]=F _(g) [k]+F _(a) [k]  (31)where F_(g)[k] is a force applied to the ball by gravity, and F_(a)[k]is a force generated when the ball changes its direction while moving.

FIG. 5 is a view for explaining power components that act on the ball.

In FIG. 5, a force of gravity that pulls down the ball is denoted by mg,and the component applied in the w direction to the ball, amongcomponents of the force mg of gravity, is denoted by F_(g)[k].

Here, considering the fact that the direction of the force applied tothe ball is in the opposite direction of the gradient vector, F_(g)[k]can be expressed using the characteristics of trigonometric functions ofequation 32, as equation 33 below.

$\begin{matrix}{{\sin\;\theta\;\cos\;\theta} = {\frac{\sin\;\theta\;\cos\;\theta}{{\cos^{2}\theta} + {\sin^{2}\theta}} = {\frac{\sin\;{\theta/\cos}\;\theta}{1 + {\sin^{2}{\theta/\cos^{2}}\theta}} = \frac{\tan\;\theta}{1 + {\tan^{2}\theta}}}}} & (32) \\{{F_{g}\lbrack k\rbrack} = {{{- {mg}}\;\sin\;\theta_{k}\cos\;\theta_{k}} = {{- {mg}}\frac{\tan\;\theta_{k}}{1 + {\tan^{2}\theta_{k}}}}}} & (33)\end{matrix}$

By applying tan θ_(k)=∇[k] to equation 33, equation 33 is rewritten toequation 34.

$\begin{matrix}{{F_{g}\lbrack k\rbrack} = {{- {mg}}\frac{\nabla\lbrack k\rbrack}{1 + {\nabla\lbrack k\rbrack^{2}}}}} & (34)\end{matrix}$

Then, a process of obtaining F_(a) will be described. When adifferentiable function y=f(x) is given, curvature κ can be expressed asequation 35 below.

$\begin{matrix}{\kappa = \frac{y^{''}}{\left( {1 + y^{\prime\; 2}} \right)^{3/2}}} & (35)\end{matrix}$

Acceleration generated when a certain object has a curvilinear motion ata velocity of v is in a direction perpendicular to the concave part ofthe curve, and if curvature of the curve is κ, the magnitude of theacceleration a can be expressed as equation 36 below.a=v ²κ  (36)

When a ball goes down or up along the inner inclined surface of acontainer as illustrated in FIG. 4, the movement velocity v of the ballcan be written by the horizontal velocity v[k] (that is, a component ofthe w[k] axis) of the ball, as equation 37 below.

$\begin{matrix}{v = \frac{v\lbrack k\rbrack}{\cos\;\theta_{k}}} & (37)\end{matrix}$

FIGS. 6A through 6C are views for explaining a change in accelerationaccording to a change in movement direction of a ball.

FIG. 6A shows the relationship between v and v[k]. Accordingly, theacceleration a can be written as equation 38 below.

$\begin{matrix}{a = {\frac{{v\lbrack k\rbrack}^{2}}{\cos^{2}\theta_{k}}{\kappa.}}} & (38)\end{matrix}$

As illustrated in FIG. 6B, acceleration a is in a directionperpendicular to the inner surface of a container, and a_(w) which is acomponent of the w[k] direction among the acceleration a can be writtenby equation 39 below.

$\begin{matrix}{a_{w} = {{a\;\sin\;\theta_{k}} = {{\frac{{v\lbrack k\rbrack}^{2}\sin\;\theta_{k}}{\cos^{2}\theta_{k}}\kappa} = {\frac{{v\lbrack k\rbrack}^{2}\tan\;\theta_{k}}{\cos\;\theta_{k}}\kappa}}}} & (39)\end{matrix}$

As seen from FIG. 6C, since tan θ_(k)=y′ and

${{\cos\;\theta_{k}} = \frac{1}{\sqrt{{\tan^{2}\theta_{k}} + 1}}},$equation 40 is induced, as follows.

$\begin{matrix}{a_{w} = {{{v\lbrack k\rbrack}^{2}y^{\prime}\sqrt{y^{\prime\; 2} + 1}\kappa} = {{{v\lbrack k\rbrack}^{2}y^{\prime}\sqrt{y^{\prime\; 2} + 1}\frac{y^{''}}{\left( {1 + y^{\prime\; 2}} \right)^{3/2}}} = {{v\lbrack k\rbrack}^{2}y^{\prime}\frac{y^{''}}{1 + y^{\prime\; 2}}}}}} & (40)\end{matrix}$

If y′=∇[k] and

$y^{''} = {\frac{\mathbb{d}^{2}ɛ}{\mathbb{d}{w\lbrack k\rbrack}^{2}}.}$are applied to equation 40, equation 41 is induced from|F_(a)[k]|=ma_(w)

$\begin{matrix}{{F_{a}\lbrack k\rbrack} = {{- {{mv}\lbrack k\rbrack}^{2}}{\nabla\lbrack k\rbrack}\frac{y^{''}}{1 + {\nabla\lbrack k\rbrack^{2}}}}} & (41)\end{matrix}$

Finally, a 1-dimensional gravity search adaptive algorithm induced fromequations 29, 34, and 41 is expressed by equation 42 below.

$\begin{matrix}{{\begin{bmatrix}{w\left\lbrack {k + 1} \right\rbrack} & {v\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{w\lbrack k\rbrack} & {v\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{g\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}},} & (42)\end{matrix}$where

${\beta = {1 - \frac{f}{m}}},$m is the mass of the ball, f is the friction coefficient of liquid, andg is the gravitational acceleration. g[k] is defined by equation 43below.

$\begin{matrix}{{g\lbrack k\rbrack} = {{{- g}\frac{\nabla\lbrack k\rbrack}{1 + {\nabla\lbrack k\rbrack^{2}}}} - {{v\lbrack k\rbrack}^{2}{\nabla\lbrack k\rbrack}\frac{y^{''}}{1 + {\nabla\lbrack k\rbrack^{2}}}}}} & (43)\end{matrix}$

By applying equation 18 to equation 43, equation 44 is deduced asfollows.

$\begin{matrix}{{g\lbrack k\rbrack} = {{{- g}\frac{{2\sigma_{x}^{2}{w\lbrack k\rbrack}} - {2\; p}}{1 + \left( {{2\;\sigma_{x}^{2}{w\lbrack k\rbrack}} - {2\; p}} \right)^{2}}} - \frac{2\;\sigma_{x}^{2}{v\lbrack k\rbrack}^{2}\left( {{2\;\sigma_{x}^{2}{w\lbrack k\rbrack}} - {2\; p}} \right)}{1 + \left( {{2\;\sigma_{x}^{2}{w\lbrack k\rbrack}} - {2\; p}} \right)^{2}}}} & (44)\end{matrix}$

<N-Dimensional Gravity Search Adaptive Algorithm>

The 1-dimensional gravity search adaptive algorithm can be easilyexpanded to an N-dimensional gravity search adaptive algorithm. It isassumed that {right arrow over (x)}[k] and {right arrow over (w)}[k] areN-dimensional signal and coefficient vectors, respectively.

FIG. 7A shows an N-dimensional gradient vector {right arrow over (∇)}[k]when N=2.

Since {right arrow over (F)}_(g)[k] indicates the opposite direction ofthe N-dimensional gradient vector {right arrow over (∇)}[k], equation 45is deduced from equation 33.

$\begin{matrix}{{{\overset{\rightharpoonup}{F}}_{g}\lbrack k\rbrack} = {{- {mg}}\frac{\tan\;\theta_{k}}{1 + {\tan^{2}\theta_{k}}}\frac{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}}} & (45)\end{matrix}$At this time, since

${{\Delta\; y} = {{\frac{\partial y}{\partial w_{1}}\Delta\; w_{1}} + {\frac{\partial y}{\partial w_{2}}\Delta\; w_{2}} + \ldots + {\frac{\partial y}{\partial w_{N}}\Delta\; w_{N}}}},$tan θ_(k) can be expressed as equation 46 below.

$\begin{matrix}{{\tan\;\theta_{k}} = {\frac{\Delta\; y}{\sqrt{{\Delta\; w_{1}^{2}} + {\Delta\; w_{2}^{2}} + \ldots + {\Delta\; w_{N}^{2}}}} = \frac{\begin{bmatrix}{\Delta\; w_{1}} & {\Delta\; w_{2}} & \ldots & {\Delta\; w_{N}}\end{bmatrix}\begin{bmatrix}\frac{\partial y}{\partial w_{1}} \\\frac{\partial y}{\partial w_{2}} \\\vdots \\\frac{\partial y}{\partial w_{N}}\end{bmatrix}}{\sqrt{\begin{bmatrix}{\Delta\; w_{1}} & {\Delta\; w_{2}} & \ldots & {\Delta\; w_{N}}\end{bmatrix}\begin{bmatrix}{\Delta\; w_{1}} \\{\Delta\; w_{2}} \\\vdots \\{\Delta\; w_{N}}\end{bmatrix}}}}} & (46)\end{matrix}$

Referring to FIG. 7A, equation 47 can be deduced for an arbitrary ε whenε<<1, as follows.

$\begin{matrix}{\begin{bmatrix}{\Delta\; w_{1}} \\{\Delta\; w_{2}} \\\vdots \\{\Delta\; w_{N}}\end{bmatrix} = {\varepsilon\frac{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}}} & (47)\end{matrix}$

Here, since

${{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack} = \begin{bmatrix}\frac{\partial y}{\partial w_{1}} \\\frac{\partial y}{\partial w_{2}} \\\vdots \\\frac{\partial y}{\partial w_{N}}\end{bmatrix}},$tan θ_(k) can be expressed as equation 48 below.

$\begin{matrix}{{\tan\;\theta_{k}} = {\frac{\varepsilon\frac{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}}{\sqrt{\varepsilon^{2}\frac{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}} = \sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}} & (48)\end{matrix}$

Accordingly, in the case of the N-dimensional gravity search adaptivealgorithm, {right arrow over (F)}_(g)[k] can be represented by equation49 below.

$\begin{matrix}{{{\overset{\rightharpoonup}{F}}_{g}\lbrack k\rbrack} = {{{- {mg}}\frac{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}\frac{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}} = {{- {mg}}\frac{1}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}} & (49)\end{matrix}$

In order to calculate

_(a)[k], curvature of the container in the direction of the movement ofthe ball has to be obtained.

FIG. 7B shows an N-dimensional velocity vector {right arrow over (v)}[k]when N=2.

The direction of the N-dimensional velocity vector {right arrow over(v)}[k] is generally not parallel to the direction of {right arrow over(∇)}[k].

Since

${{\Delta\; y} = {{\frac{\partial y}{\partial w_{1}}\Delta\; w_{1}} + {\frac{\partial y}{\partial w_{2}}\Delta\; w_{2}} + \ldots + {\frac{\partial y}{\partial w_{N}}\Delta\; w_{N}}}},$equation 50 can be obtained.

$\begin{matrix}{{\tan\;\phi_{k}} = {\frac{\Delta\; y}{\sqrt{{\Delta\; w_{1}^{2}} + {\Delta\; w_{2}^{2}} + \ldots + {\Delta\; w_{N}^{2}}}} = \frac{\begin{bmatrix}{\Delta\; w_{1}} & {\Delta\; w_{2}} & \ldots & {\Delta\; w_{N}}\end{bmatrix}\begin{bmatrix}\frac{\partial y}{\partial w_{1}} \\\frac{\partial y}{\partial w_{2}} \\\vdots \\\frac{\partial y}{\partial w_{N}}\end{bmatrix}}{\sqrt{\begin{bmatrix}{\Delta\; w_{1}} & {\Delta\; w_{2}} & \ldots & {\Delta\; w_{N}}\end{bmatrix}\begin{bmatrix}{\Delta\; w_{1}} \\{\Delta\; w_{2}} \\\vdots \\{\Delta\; w_{N}}\end{bmatrix}}}}} & (50)\end{matrix}$

However, if

$\begin{bmatrix}{\Delta\; w_{1}} \\{\Delta\; w_{2}} \\\vdots \\{\Delta\; w_{N}}\end{bmatrix} = {\varepsilon\;{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}$for an arbitrary ε when ε<<1, tan Φ_(k) can be expressed as equation 51below.

$\begin{matrix}{{\tan\;\phi_{k}} = {\frac{\varepsilon\;{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\sqrt{\varepsilon^{2}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} = \frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}}} & (51)\end{matrix}$

If s is a trajectory of the ball that has moved along the inner surfaceof the container, curvature is defined as a change rate in movementdirection when the ball moves by a unit distance. An example of suchcurvature is shown in FIG. 8. Referring to FIG. 8, as the ball moves byΔs, the movement direction changes by ΔΦ. Accordingly, bydifferentiating both sides of equation 51 with respect to s, thecurvature is defined by equation 52 below.

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}s}\tan\;\phi_{k}} = {{\frac{\mathbb{d}}{\mathbb{d}s}\left( \frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} \right)} = {{\frac{1}{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}\frac{\mathbb{d}}{\mathbb{d}s}\left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)} = {\frac{1}{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}\frac{\mathbb{d}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\mathbb{d}s}}}}} & (52)\end{matrix}$

Here, equation 52 is arranged to equation 53 below.

$\begin{matrix}\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}s}\tan\;\phi_{k}} = {\frac{\mathbb{d}}{\mathbb{d}s}\left( \frac{\sin\;\phi_{k}}{\cos\;\phi_{k}} \right)}} \\{= {\frac{{\sin^{2}\phi_{k}} + {\cos^{2}\phi_{k}}}{\cos^{2}\phi_{k}}\frac{\mathbb{d}\phi_{k}}{\mathbb{d}s}}} \\{= {\frac{1}{\cos^{2}\phi_{k}}\frac{\mathbb{d}\phi_{k}}{\mathbb{d}s}}} \\{= {\frac{1}{\cos^{2}\phi_{k}}k}}\end{matrix} & (53)\end{matrix}$

Accordingly, equation 54 can be obtained from equation 53.

$\begin{matrix}{\kappa = {\cos^{2}\phi_{k}\frac{1}{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}\frac{\mathbb{d}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\mathbb{d}s}}} & (54)\end{matrix}$

Here,

$\frac{\mathbb{d}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\mathbb{d}s}.$can be represented as a product of R and a vector, as shown in equation55.

$\begin{matrix}\begin{matrix}{\frac{\mathbb{d}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\mathbb{d}s} = {{\frac{\partial w_{1}}{\partial s}\frac{\partial{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\partial w_{1}}} + {\frac{\partial w_{2}}{\partial s}\frac{\partial{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\partial w_{2}}} + \ldots + {\frac{\partial w_{N}}{\partial s}\frac{\partial{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\partial w_{N}}}}} \\{= {{\begin{bmatrix}\frac{\partial{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\partial w_{1}} & \frac{\partial{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\partial w_{2}} & \ldots & \frac{\partial{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\partial w_{N}}\end{bmatrix}\begin{bmatrix}\frac{\partial w_{1}}{\partial s} \\\frac{\partial w_{2}}{\partial s} \\\vdots \\\frac{\partial w_{N}}{\partial s}\end{bmatrix}} = {2\;{R\begin{bmatrix}\frac{\partial w_{1}}{\partial s} \\\frac{\partial w_{2}}{\partial s} \\\vdots \\\frac{\partial w_{N}}{\partial s}\end{bmatrix}}}}}\end{matrix} & (55)\end{matrix}$

Here, since

$\begin{matrix}{{{\Delta\; s} = {\sqrt{{\varepsilon^{2}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {\varepsilon{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}} = {\varepsilon\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}}},{\frac{\partial w_{n}}{\partial s} = {{\lim\limits_{\varepsilon->0}\frac{\Delta\; w_{n}}{\Delta\; s}} = {{\lim\limits_{\varepsilon->0}\frac{\varepsilon\;{v_{n}\lbrack k\rbrack}}{\varepsilon\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}} = {\frac{v_{n}\lbrack k\rbrack}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}.}}}}} & (56)\end{matrix}$

Where, v_(n)[k] is the nth element of the vector {right arrow over(v)}[k]. Accordingly, equation 57 is deduced as follows.

$\begin{matrix}{\begin{bmatrix}\frac{\partial w_{1}}{\partial s} \\\frac{\partial w_{2}}{\partial s} \\\vdots \\\frac{\partial w_{N}}{\partial s}\end{bmatrix} = {{\frac{1}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}\begin{bmatrix}{v_{0}\lbrack k\rbrack} \\{v_{1}\lbrack k\rbrack} \\\vdots \\{v_{N - 1}\lbrack k\rbrack}\end{bmatrix}} = {\frac{1}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}} & (57)\end{matrix}$

Accordingly, equation 58 is deduced from equation 57, as follows.

$\begin{matrix}{\frac{\mathbb{d}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}{\mathbb{d}s} = {\frac{1}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}2\; R{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} & (58)\end{matrix}$

As a result, the curvature is given as equation 59 below.

$\begin{matrix}{\kappa = {\cos^{2}\phi_{k}\frac{1}{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}\frac{1}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}2\; R{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} & (59)\end{matrix}$

Accordingly, the acceleration a can be obtained by equation 60 below.

$\begin{matrix}{a = {{v^{2}\kappa} = {{\frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}{\cos^{2}\phi_{k}}\kappa} = {\frac{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}2\; R\;{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}}} & (60)\end{matrix}$Since the acceleration a is always in a direction perpendicular to thesurface and is not related to the direction of {right arrow over(v)}[k], {right arrow over (a)}[k] can be expressed by equation 61below. That is, the direction of a is perpendicular to the surface, andthe direction of {right arrow over (v)}[k] is tangential to the surface,but can be in any direction.

$\begin{matrix}\begin{matrix}{{\overset{\rightharpoonup}{a}\lbrack k\rbrack} = {{- a}{{\sin\;\phi_{k}}}\frac{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}}} \\{= {{- {{\sin\;\phi_{k}}}}\frac{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}2\; R\;{{\overset{\rightharpoonup}{v}\lbrack k\rbrack} \cdot}}} \\{\frac{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}}\end{matrix} & (61)\end{matrix}$

At this time, since equation 62 is satisfied, equation 63 is deduced.

$\begin{matrix}{{{\sin\;\phi_{k}}} = {\frac{{\tan\;\phi_{k}}}{\sqrt{1 + {\tan^{2}\phi_{k}}}} = {\frac{\frac{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}}{\sqrt{1 + \frac{\left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}} = \frac{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}}}} & (62) \\\begin{matrix}{\mspace{79mu}{{\overset{\rightharpoonup}{a}\lbrack k\rbrack} = {- \frac{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}}}}} \\{{\frac{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{\sqrt{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}}} \cdot {\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}}2\; R{{\overset{\rightharpoonup}{v}\lbrack k\rbrack} \cdot}} \\{\frac{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}} \\{= {{- 2}{\frac{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}} \cdot {\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}}R{{\overset{\rightharpoonup}{v}\lbrack k\rbrack} \cdot}}} \\{\frac{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}}\end{matrix} & (63)\end{matrix}$

Finally, the N-dimensional gravity search adaptive algorithm is arrangedto equation 64 below.

$\begin{matrix}{{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}},} & (64)\end{matrix}$where

${\beta = {1 - \frac{f}{m}}},$where m is the mass of the ball, f is the friction coefficient of theliquid, and g is the gravitational acceleration. Also, g[k] is definedas equation 65 below.

$\begin{matrix}{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {{\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} - {2{\frac{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}} \cdot {\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}}R\;{{\overset{\rightharpoonup}{v}\lbrack k\rbrack} \cdot \frac{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}}}}} & (65)\end{matrix}$

<N-dimensional Gravity Search Adaptive Algorithm Having ComplexCoefficients>

If {right arrow over (x)}[k] is an N-dimensional complex signal vectorand {right arrow over (w)}[k] is an N-dimensional complex coefficientvector, the cost function given by equation 3 above can be rewritten toequation 66 below.

$\begin{matrix}{{ɛ = {\sigma_{y}^{2} - {{\overset{\rightharpoonup}{p}}^{H}{\overset{\rightharpoonup}{w}\lbrack k\rbrack}} - {{\overset{\rightharpoonup}{w}\lbrack k\rbrack}^{H}\overset{\rightharpoonup}{p}} + {{\overset{\rightharpoonup}{w}\lbrack k\rbrack}^{H}R\;{\overset{\rightharpoonup}{w}\lbrack k\rbrack}} + \sigma_{n}^{2}}},} & (66)\end{matrix}$

Real vectors

_(r);

_(i);

_(r)[k],

_(i)[k] and real matrices R_(r) and R_(i) are defined by equations 67,68, and 69, respectively, below.

=

_(r) +j

_(i)  (67)

[k]=

_(r) [k]+j

_(i) [k]  (68)R=R _(r) +jR _(i)  (69)

Here, the real matrices R_(r) and R_(i) satisfy equation 70 below.R _(r) ^(T) =R _(r) R _(i) ^(T) =−R _(i)  (70)

Equations 67, 68, and 69 are applied to equation 66, and equation 66 isrearranged to obtain equation 71 below.ε=σ_(y) ²−2

[k] ^(T)

+

[k] ^(T) R

[k]+σ _(n) ²  (71)

Here, {right arrow over (p)}, {right arrow over (w)}[k], and R aredefined by equation 72 below.

$\begin{matrix}{{\overset{\rightharpoonup}{P} = \begin{bmatrix}{\overset{\rightharpoonup}{p}}_{r} \\{\overset{\rightharpoonup}{p}}_{i}\end{bmatrix}};{{\overset{\rightharpoonup}{W}\lbrack k\rbrack} = \begin{bmatrix}{{\overset{\rightharpoonup}{w}}_{r}\lbrack k\rbrack} \\{{\overset{\rightharpoonup}{w}}_{i}\lbrack k\rbrack}\end{bmatrix}};{\mathcal{R} = \begin{bmatrix}R_{r} & {- R_{i}} \\R_{i} & R_{r}\end{bmatrix}}} & (72)\end{matrix}$

Equations 71 and 72 represent a problem with 2N-dimensional realcoefficients. Accordingly, the gravity search adaptive algorithm usingreal numbers as coefficients, as obtained in equations 64 and 65, isused to calculate a 2N-dimensional gravity search adaptive algorithm asexpressed as equations 73 and 74 below.

$\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{W}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{V}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{W}\lbrack k\rbrack} & {\overset{\rightharpoonup}{V}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{G}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (73) \\{{{\overset{\rightharpoonup}{G}\lbrack k\rbrack} = {{\frac{- g}{1 + {{\overset{\rightharpoonup}{D}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{D}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{D}\lbrack k\rbrack}} - {2{\frac{{{{\overset{\rightharpoonup}{V}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{D}\lbrack k\rbrack}}} \cdot \sqrt{{\overset{\rightharpoonup}{V}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{V}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{V}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{V}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{V}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{D}\lbrack k\rbrack}} \right)^{2}} \cdot {\overset{\rightharpoonup}{V}\lbrack k\rbrack}^{T}}\mathcal{R}\;{{\overset{\rightharpoonup}{V}\lbrack k\rbrack} \cdot \frac{\overset{\rightharpoonup}{D}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{D}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{D}\lbrack k\rbrack}}}}}}},\mspace{20mu}{{{where}\mspace{79mu}{\overset{\rightharpoonup}{D}\lbrack k\rbrack}} = {\frac{\partial ɛ}{\partial{\overset{\rightharpoonup}{W}\lbrack k\rbrack}} = {{{- 2}\overset{\rightharpoonup}{P}} + {2\mathcal{R}\;{{\overset{\rightharpoonup}{W}\lbrack k\rbrack}.}}}}}} & (74)\end{matrix}$

It will be seen from equation 72 that equations 73 and 74 are differentexpressions of an adaptive algorithm for obtaining {right arrow over(w_(r))}[k] and {right arrow over (w_(i))}[k].

Accordingly, after obtaining {right arrow over (w_(r))}[k] and {rightarrow over (w_(i))}[k] from the adaptive algorithm of equations 73 and74, equations 67, 68, and 69 are used to obtain an N-dimensional complexgravity search adaptive algorithm, as expressed as equations 75 and 76below.

$\begin{matrix}{\mspace{85mu}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}}} & (75) \\{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {{\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} - {\frac{{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + {\frac{1}{4}\left( {{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} \right)^{2}}} \cdot \frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}R\;{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}} & (76)\end{matrix}$

<Simplification of N-Dimensional Gravity Search Adaptive Algorithm>

{right arrow over (g)}[k] in equation 76 is composed of two terms; thefirst term is generated when gravity is applied to the ball, and thesecond term is generated by acceleration when the ball changes itsmovement direction while moving along the curved surface of thecontainer.

FIG. 9 is a view for explaining a change in acceleration when themovement direction of a ball changes.

Since a change in acceleration due to a change in movement direction ofa ball has a very small magnitude compared to the gravitationalacceleration, for simplification of implementation, the acceleration dueto the change in movement direction is ignored since it has littleinfluence on performance of the adaptive algorithm. If the second termof {right arrow over (g)}[k] in equation 76 is ignored, a simplifiedgravity search adaptive algorithm is deduced, which is expressed byequation 77 below.

$\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (77) \\{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} & (78)\end{matrix}$

In equations 77 and 78, β is a parameter associated with inertia. If βis large, that is, if β is close to 1, the velocity of the ball does notchange easily since the ball has a strong tendency to maintaining itsown velocity, while if β is small, that is, if β is close to 0, the ballhas a weak tendency to maintain its own velocity. If β is zero, thevelocity of the ball depends only on a force currently applied to theball regardless of the previous velocity of the ball. That is, when β iszero, the adaptive algorithm of equations 75 and 76 is rewritten toequation 79 below.

$\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} - {\frac{{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + {\frac{1}{4}\left( {{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} \right)^{2}}} \cdot \frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}R\;{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}} & (79)\end{matrix}$

Also, when β is zero, the adaptive algorithm of equations 77 and 78 isrewritten to equation 81 below.

$\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}} & (80)\end{matrix}$

<Gravity Search LMS (GS-LMS) Algorithm>

Like the steepest descent algorithm, in order to implement a GS-LMSalgorithm, R and {right arrow over (p)} have to be given. Like wheninducing a LMS algorithm, the case of estimating R and {right arrow over(p)} from instantaneous values of the signal as given in equations 14and 15 is called a GS-LMS algorithm.

A gradient vector can be deduced from equations 14 and 15, which isexpressed as equation 81 below.

$\begin{matrix}\begin{matrix}{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack} = {{{{- 2}\;\overset{\rightharpoonup}{p}} + {2\; R\;{\overset{\rightharpoonup}{w}\lbrack k\rbrack}}} = {{- 2}\left( {{{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{y^{*}\lbrack k\rbrack}} - {{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{w}\lbrack k\rbrack}}} \right)}}} \\{= {{{- 2}\;{\overset{\rightharpoonup}{x}\lbrack k\rbrack}\left( {{y^{*}\lbrack k\rbrack} - {{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{w}\lbrack k\rbrack}}} \right)} = {{- 2}\;{\overset{\rightharpoonup}{x}\lbrack k\rbrack}\left( {{y^{*}\lbrack k\rbrack} - {{\hat{y}}^{*}\lbrack k\rbrack}} \right)}}} \\{= {{- 2}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}\end{matrix} & (81)\end{matrix}$

Equation 81 is applied to equation 76, thus deducing equation 82.

$\begin{matrix}\begin{matrix}{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {{\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} -}} \\{\frac{{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + {\frac{1}{4}\left( {{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} \right)^{2}}} \cdot} \\{\frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}R\;{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \\{= {{\frac{2g}{1 + {1\;{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}e*\lbrack k\rbrack}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} +}} \\{\frac{2{{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} + {{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} + {{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} \right)^{2}} \cdot} \\{\frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}{2\sqrt{{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}} \cdot \left( {2\;{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} \right)}\end{matrix} & (82)\end{matrix}$

If it is defined that z[k]=

[k]^(H)

[k] and {right arrow over (g)}[k] is represented in terms of z[k],equation 82 can be rewritten to equation 83 below.

$\begin{matrix}\begin{matrix}{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {{\frac{2g}{1 + {4\;{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}e*\lbrack k\rbrack}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} +}} \\{\frac{2{{{{{z\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} + {{e\lbrack k\rbrack}{z^{*}\lbrack k\rbrack}}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{{z\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} + {{e\lbrack k\rbrack}{z^{*}\lbrack k\rbrack}}} \right)^{2}} \cdot} \\{\frac{{z\lbrack k\rbrack}{z^{*}\lbrack k\rbrack}}{\sqrt{{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} \\{= {{\frac{2g}{1 + {4\;{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} +}} \\{\frac{4{{{\Re\left\{ {{z\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} \right\}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + {4\;\Re\left\{ {{z\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} \right\}^{2}}} \cdot} \\{{\frac{{{z\lbrack k\rbrack}}^{2}}{\sqrt{{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}},}\end{matrix} & (83)\end{matrix}$

Here,

{•} represents the real part of the complex number. Accordingly, fromequations 83 and 73, the GS-LMS algorithm is given as equations 84 and85 below.

$\begin{matrix}{\mspace{79mu}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}}} & (84) \\{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\left( {\frac{2g}{1 + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}} + {\frac{4{{{\left\{ {{z\lbrack k\rbrack}e^{*}} \right\}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + {4\left\{ {{z\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} \right\}^{2}}} \cdot \frac{{{z\lbrack k\rbrack}}^{2}}{\sqrt{{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}}} \right){\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}} & (85)\end{matrix}$

Meanwhile, if the acceleration component due to a change in movementdirection of the ball is ignored from {right arrow over (g)}[k],equations 75 and 79 are rewritten to a simplified GS-LMS algorithm,which is expressed as equations 86 and 87 below.

$\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (86) \\{{g\lbrack k\rbrack} = {\frac{2g}{1 + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}} & (87)\end{matrix}$

If β is zero, equation 79 can be rewritten to a simplified GS-LMSalgorithm, which is expressed as equation 88 below.

$\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\left( {\frac{2g}{1 + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}} + {\frac{4{{{\left\{ {{z\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} \right\}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + {4\left\{ {{z\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}} \right\}^{2}}} \cdot \frac{{{z\lbrack k\rbrack}}^{2}}{\sqrt{{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}}} \right){\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}} & (88)\end{matrix}$

If β is zero, equation 80 can be rewritten to a simplified GS-LMSalgorithm, which is expressed as equation 89 below.

$\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\frac{2g}{1 + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}} & (89)\end{matrix}$

In equation 87, {right arrow over (g)}[k] has been normalized withrespect to both {right arrow over (x)}[k] and {right arrow over (e)}[k].Accordingly, the convergence speed of the simplified GS-LMS algorithm ofequations 86 and 87 may become slow during the early stage of adaptationwhen the power of e[k] is large. In this case, one method for improvingthe convergence speed is to vary g over time.

For example, it is possible to make g be in proportion to the power ofan error signal e[k], as follows.g[k]={right arrow over (g)}|e[k]| ²  (90)

By applying equation 90 to equation 87, equations 91 and 92 are obtainedas follows.

$\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (91) \\{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\frac{2\overset{\sim}{g}{{e\lbrack k\rbrack}}^{2}}{1 + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}} & (92)\end{matrix}$

Here, a range of the {tilde over (g)} value is given by equation 93below.

$\begin{matrix}{0 < \frac{\overset{\sim}{g}}{\left( {1 - \beta} \right)} < 2} & (93)\end{matrix}$

If β is zero, equations 91 and 92 can be rewritten to equation 94 below.

$\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\frac{2\overset{\sim}{g}{{e\lbrack k\rbrack}}^{2}}{1 + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}} & (94)\end{matrix}$

The adaptive algorithm written as equation 94 has a structure that isvery similar to the normalized LMS algorithm of equation 95.

$\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\frac{\overset{\sim}{\mu}}{\delta + {{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}} & (95)\end{matrix}$

For easy comparison, equation 94 is rearranged to equation 96 below.

$\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\frac{\overset{\sim}{g}/2}{{{1/4}{{e\lbrack k\rbrack}}^{2}} + {{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}} & (96)\end{matrix}$

If equations 97 and 98 below are satisfied, equation 95 becomesidentical to equation 96.

$\begin{matrix}{\delta = \frac{1}{4{{e\lbrack k\rbrack}}^{2}}} & (97) \\{\overset{\sim}{\mu} = \frac{\overset{\sim}{g}}{2}} & (98)\end{matrix}$

This means that if there is no memory in the GS-LMS algorithm ofequation 94, the GS-LMS algorithm becomes a normalized LMS algorithm.According to a different interpretation, this means that a GS-LMSalgorithm is an enhanced version of the normalized LMS by introducingmemory or inertia.

The above interpretation about the GS-LMS algorithm has very importantsignificance. The reason is because the characteristics of the GS-LMSalgorithm can be understood based on the well-known characteristics ofthe normalized LMS algorithm.

It is assumed that equations 99 and 100 below are satisfied.1<<4e[k]

[k] ^(H)

[k]e*[k]  (99)4δ|e[k]| ²<<4e[k]

[k] ^(H)

[k]e*[k]  (100)

In this case, the GS-LMS algorithm may be further simplified. That is,under the assumption, equation 92 can be rewritten to equation 101below.

$\begin{matrix}{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\frac{2\overset{\sim}{g}{{e\lbrack k\rbrack}}^{2}}{{4\delta{{e\lbrack k\rbrack}}^{2}} + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}} & (101)\end{matrix}$

By arranging equation 101, the algorithms of equations 91 and 92 aresimplified to equation 102 and 103 below.

$\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (102) \\{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\frac{\overset{\sim}{\mu}}{\delta + {{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}} & (103)\end{matrix}$

Also, the algorithms of equations 101 and 103 become identical to thenormalized LMS algorithm when β is zero.

FIG. 10 is a diagram illustrating an example of a gravity searchadaptive apparatus.

The gravity search adaptive apparatus illustrated in FIG. 10 correspondsto an example where the GS-LMS algorithm is used. In FIG. 10, a modelingparameter calculator 1030 uses the gravity search adaptive algorithmdescribed above to update a parameter for making an error signal outputfrom an error detector 1031 converge on a minimum value and to apply theupdated parameter to a system modeling unit 1020.

FIG. 11 is a flowchart illustrating an example of a system modelingmethod using a gravity search adaptive algorithm.

Referring to FIGS. 10 and 11, an adaptive controller 1030 calculates asystem modeling parameter using the gravity search adaptive algorithm(1110). The system modeling parameter is a value for minimizing an errorsignal which is a difference between an output signal of a real systemand a modeled signal.

Then, the adaptive controller 1030 applies the system modeling parameterto the system modeling unit 1020 (1120). Then, the adaptive controller1030 applies the same signal to both the system 1010 and the systemmodeling unit 1020 (1130). Then, the adaptive controller 1030 calculatesan error signal which is a difference between an output signal of thesystem 1010 and an output signal of the system modeling unit 1020(1140). Successively, the adaptive controller 1030 compares the errorsignal to the previously calculated error signal to thereby determinewhether the error signal converges on a minimum value (1150).

If it is determined in operation 1150 that the error signal does notconverge on a minimum value, the adaptive controller 1030 returns tooperation 1110 to update the system modeling parameter using the gravitysearch adaptive algorithm. That is, the procedure from operation 1110 tooperation 1150 is repeated until the error signal converges on theminimum value.

Also, when calculating a system parameter using the gravity searchadaptive algorithm, under an assumption that a ball moves along theinner surface of a parabolic, concave container filled up with liquid,the adaptive controller 1030 sets the mass m of the ball, a frictionforce g of liquid, a sampling interval T_(s), and a force F_(k) appliedto the ball, and applies the set values to equation 37 to therebycalculate or update a parameter of the system modeling unit 1020.

Meanwhile, if it is determined in operation 1150 that the error signalconverges on the minimum, the adaptive controller 1030 estimates thecharacteristics of the system 1010 using a final parameter calculated inoperation 1160.

Therefore, as described above, the gravity search adaptive apparatus andmethod can easily adjust a convergence speed and a convergence pattern.

A number of examples have been described above. Nevertheless, it will beunderstood that various modifications may be made. For example, suitableresults may be achieved if the described techniques are performed in adifferent order and/or if components in a described system,architecture, device, or circuit are combined in a different mannerand/or replaced or supplemented by other components or theirequivalents. Accordingly, other implementations are within the scope ofthe following claims.

What is claimed is:
 1. A gravity search adaptive apparatus for modelinga system, the apparatus comprising: a system modeling circuit configuredto receive an input signal input to the system, to convert the inputsignal using a model parameter vector, and to output the convertedsignal; and an adaptive controller configured to calculate, using agravity search adaptive algorithm for solving a problem of finding aglobal minimum of a cost function which is in the form of a quadraticfunction, the model parameter vector when an error signal, which is adifference between an output signal of the system and an output signalof the system modeling circuit, converges on a minimum power value, andto apply the calculated model parameter vector to the system modelingcircuit, wherein the adaptive controller calculates a power value ε ofthe error signal, the power value ε being expressed as equation 104below:ε=σ_(y) ²−2pw[k]+σ _(x) ² w[k] ²  (104), where w[k] is the modelparameter vector resulting from k-th iteration of the calculation of themodel parameter vector, p=E{x[k]y[k]}, σ_(x) ²=E{x²[k]}, x[k] is theinput signal, y[k] is the output signal of the system, σ_(y) ² isE{y²[k]}, and E{•} represents an ensemble average.
 2. The gravity searchadaptive apparatus of claim 1, wherein the adaptive controllercomprises: an error detector configured to detect and output the errorsignal; and a modeling parameter calculator configured to calculate themodel parameter vector when the error signal converges on the minimumpower value and to apply the calculated model parameter vector to thesystem modeling unit.
 3. The gravity search adaptive apparatus of claim1, wherein the gravity search adaptive algorithm is estimated under anassumption that a ball moves along an inner surface of a parabolic,concave container filled up with liquid.
 4. The gravity search adaptiveapparatus of claim 3, wherein the gravity search adaptive algorithm usesa second-order differential equation 105 below:m{umlaut over (w)}+f{dot over (w)}=F  (105), where w represents themodel parameter vector of the system modeling circuit, F represents aforce applied to the ball in a direction of w, m represents a mass ofthe ball, f represents a friction force depending on viscosity of theliquid, and {dot over (w)} and {umlaut over (w)} represent a first-orderderivative and a second-order derivative with respect to time,respectively.
 5. The gravity search adaptive apparatus of claim 3,wherein the adaptive controller updates the model parameter vector ofthe system modeling circuit using a first-order gravity search adaptivealgorithm that is expressed as equation 106 below: $\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (106)\end{matrix}$ where ${\beta = {1 - \frac{f}{m}}},$ m represents a massof the ball, f represents a friction coefficient of the liquid, v[k] isa horizontal velocity of the ball, g represents gravitationalacceleration, g[k] is defined as equation 107 below: $\begin{matrix}{{{g\lbrack k\rbrack} = {{{- g}\;\frac{\Delta\lbrack k\rbrack}{1 + {\nabla\lbrack k\rbrack^{2\;}}}} - {{v\lbrack k\rbrack}^{2}{\nabla\lbrack k\rbrack}\frac{y^{''}}{1 + {\nabla\lbrack k\rbrack^{2}}}}}},} & (107)\end{matrix}$ where ∇[k] is defined as follows:${\frac{\mathbb{d}ɛ}{\mathbb{d}{w\lbrack k\rbrack}} = {\nabla\lbrack k\rbrack}},$and y″ is defined as follows:$y^{''} = {\frac{\mathbb{d}^{2}ɛ}{\mathbb{d}{w\lbrack k\rbrack}^{2}}.}$6. The gravity search adaptive apparatus of claim 3, wherein theadaptive controller updates the model parameter vector of the systemmodeling circuit using an N-dimensional gravity search adaptivealgorithm that is expressed as equation 108 below: $\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (108)\end{matrix}$ where w represents the model parameter vector of thesystem modeling circuit, v represents a change velocity of w,${\beta = {1 - \frac{f}{m}}},$ m represents a mass of the ball, f is afriction coefficient of the liquid, g represents gravitationalacceleration, and g[k] is defined as equation 109 below: $\begin{matrix}{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {{\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} - {2\;{\frac{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + \left( {{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} \right)^{2}} \cdot {\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{T}}R{{\overset{\rightharpoonup}{v}\lbrack k\rbrack} \cdot {\frac{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{T}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}.}}}}} & (109)\end{matrix}$ where R is defined as follows:R=E{

[k]

[k] ^(H)}, H representing Hermitian Transpose, and ∇[k] is defined asfollows:$\frac{\mathbb{d}ɛ}{\mathbb{d}{w\lbrack k\rbrack}} = {{\nabla\lbrack k\rbrack}.}$7. The gravity search adaptive apparatus of claim 3, wherein theadaptive controller updates the model parameter vector of the systemmodeling circuit using an N-dimensional gravity search adaptivealgorithm having complex coefficients that are expressed as equation 110below: $\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (110)\end{matrix}$ where w represents the model parameter vector of thesystem modeling circuit, v represents a change velocity of w,${\beta = {1 - \frac{f}{m}}},$ m represents a mass of the ball, frepresents a friction coefficient of the liquid, g representsgravitational acceleration, and g[k] is defined as equation 111 below:$\begin{matrix}{{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {{\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} - {\frac{{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + {\frac{1}{4}\left( {{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} \right)^{2}}} \cdot \frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}R{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}},} & (111)\end{matrix}$ where R is defined as follows:R=E{

[k]

[k] ^(H)}, H representing Hermitian Transpose, and ∇[k] is defined asfollows:$\frac{\mathbb{d}ɛ}{\mathbb{d}{w\lbrack k\rbrack}} = {{\nabla\lbrack k\rbrack}.}$8. The gravity search adaptive apparatus of claim 7, wherein if aparameter β associated with inertia is zero, algorithms of equations 110and 111 are expressed as equation 112 below: $\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} - {\frac{{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + {\frac{1}{4}\left( {{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} \right)^{2}}} \cdot \frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}R{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}.}}}} & (112)\end{matrix}$
 9. The gravity search adaptive apparatus of claim 3,wherein the adaptive controller updates the model parameter vector ofthe system modeling circuit using a gravity search adaptive algorithmthat is expressed as equation 113 below: $\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (113)\end{matrix}$ where w represents the model parameter vector of thesystem modeling circuit, ${\beta = {1 - \frac{f}{m}}},$ v represents achange velocity of w, m represents a mass of the ball, f represents afriction coefficient of the liquid, g represents gravitationalacceleration, and g[k] is defined as equation 114 below: $\begin{matrix}{{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}},} & (114)\end{matrix}$ where ∇[k] is defined as follows:$\frac{\mathbb{d}ɛ}{\mathbb{d}{w\lbrack k\rbrack}} = {{\nabla\lbrack k\rbrack}.}$10. The gravity search adaptive apparatus of claim 9, wherein if aparameter β associated with inertia is zero, algorithms of equations 113and 114 are expressed as equation 115 below: $\begin{matrix}{{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\frac{- g}{1 + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} - {\frac{{{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}} \cdot \sqrt{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}}{{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}} + {\frac{1}{4}\left( {{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}} + {{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{v}\lbrack k\rbrack}}} \right)^{2}}} \cdot \frac{{\overset{\rightharpoonup}{v}\lbrack k\rbrack}^{H}R{\overset{\rightharpoonup}{\; v}\lbrack k\rbrack}}{\sqrt{{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack^{H}}{\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}} \cdot {\overset{\rightharpoonup}{\nabla}\lbrack k\rbrack}}}},} & (115)\end{matrix}$ where R is defined as follows:R=E{

[k]

[k] ^(H)}, H representing Hermitian Transpose.
 11. The gravity searchadaptive apparatus of claim 3, wherein the adaptive controller updatesthe model parameter vector of the system modeling circuit using agravity search adaptive algorithm that is expressed as equation 116below: $\begin{matrix}{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (116)\end{matrix}$ where w represents the model parameter vector of thesystem modeling circuit, v represents a change velocity of w,${\beta = {1 - \frac{f}{m}}},$ m represents a mass of the ball, frepresents a friction coefficient of the liquid, g representsgravitational acceleration, and g[k] is defined as equation 117 below$\begin{matrix}{{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\frac{2g}{1 + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}},} & (117)\end{matrix}$ where e[k] represents the error signal and H representsHermitian Transpose.
 12. The gravity search adaptive apparatus of claim11, wherein the adaptive controller applies equation 118 to equation117, thus obtaining equations 119 and 120 below: $\begin{matrix}{{g\lbrack k\rbrack} = {\overset{\sim}{g}{{e\lbrack k\rbrack}}^{2}}} & (118) \\{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}} & (119) \\{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\frac{2\overset{\sim}{g}{{e\lbrack k\rbrack}}^{2}}{1 + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{{e^{*}\lbrack k\rbrack}.}}} & (120)\end{matrix}$
 13. The gravity search adaptive apparatus of claim 12,wherein if β is zero, the adaptive controller defines equations 119 and120 by equation 121 below: $\begin{matrix}{{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} = {{\overset{\rightharpoonup}{w}\lbrack k\rbrack} + {\frac{2\overset{\sim}{g}{{e\lbrack k\rbrack}}^{2}}{1 + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{{e^{*}\lbrack k\rbrack}.}}}} & (121)\end{matrix}$
 14. The gravity search adaptive apparatus of claim 12,wherein when equations 122 and 123 are satisfied, the adaptivecontroller defines the equation 120 by equation 124, and obtains analgorithm defined as equations 125 and 126 from the equation 124:$\begin{matrix}{{1 ⪡ {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}},} & (122) \\{{{4\delta{{e\lbrack k\rbrack}}^{2}} ⪡ {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}},} & (123) \\{{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\frac{2\overset{\sim}{g}{{e\lbrack k\rbrack}}^{2}}{{4\delta{{e\lbrack k\rbrack}}^{2}} + {4{e\lbrack k\rbrack}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e^{*}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{e\lbrack k\rbrack}}},} & (124) \\{{\begin{bmatrix}{\overset{\rightharpoonup}{w}\left\lbrack {k + 1} \right\rbrack} & {\overset{\rightharpoonup}{v}\left\lbrack {k + 1} \right\rbrack}\end{bmatrix} = {{\begin{bmatrix}{\overset{\rightharpoonup}{w}\lbrack k\rbrack} & {\overset{\rightharpoonup}{v}\lbrack k\rbrack}\end{bmatrix}\begin{bmatrix}1 & 0 \\\beta & \beta\end{bmatrix}} + {{\overset{\rightharpoonup}{g}\lbrack k\rbrack}\begin{bmatrix}1 & 1\end{bmatrix}}}},{and}} & (125) \\{{\overset{\rightharpoonup}{g}\lbrack k\rbrack} = {\frac{\overset{\sim}{\mu}}{\delta + {{\overset{\rightharpoonup}{x}\lbrack k\rbrack}^{H}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}}}{\overset{\rightharpoonup}{x}\lbrack k\rbrack}{{e^{*}\lbrack k\rbrack}.}}} & (126)\end{matrix}$ where $\delta = {\frac{1}{4{{e\lbrack k\rbrack}}^{2}}.}$15. A method comprising: calculating, by an adaptive controller using agravity search adaptive algorithm for solving a problem of finding aglobal minimum of a cost function which is in the form of a quadraticfunction, a model parameter vector of a system modeling circuitcorresponding to a minimum power value of an error signal which is adifference between an output signal of a system and an output signal ofthe system modeling circuit when an input signal is input to the systemand the system modeling circuit; and applying, by the adaptivecontroller, the model parameter vector to the system modeling circuit,wherein the calculating further comprises: calculating a power value εof the error signal that is expressed as follows:ε=σ_(y) ²−2pw[k]+σ _(x) ² w[k] ², where w[k] is the model parametervector resulting from k-th iteration of the calculation of the modelparameter vector, p is E{x[k]y[k]}, σ_(x) ² is E{x²[k]}, x[k] is theinput signal, y[k] is the output signal of the system, σ_(y) ² isE{y²[k]}, and E{•} represents an ensemble average.
 16. The method ofclaim 15, further comprising: comparing, by the adaptive controller, theerror signal to a previously calculated error signal to determinewhether the error signal converges on the minimum power value; andupdating, by the adaptive controller, if the error signal does notconverge on the minimum power value, the model parameter vector usingthe gravity search adaptive algorithm.
 17. The method of claim 15,further comprising: comparing, by the adaptive controller, the errorsignal to a previously calculated error signal to determine whether theerror signal converges on the minimum power value; and estimating, bythe adaptive controller, if the error signal converges on the minimumpower value, system characteristics of the system using the calculatedmodel parameter vector.